Representing Numbers and Letters with Binary: Crash Course Computer Science #4

0:03 Hi I’m Carrie Anne, this is Crash Course Computer Science

0:06 and today we’re going to talk about

0:08 how computers store and represent numerical data.

0:10 Which means we’ve got to talk about Math!

0:12 But don’t worry.

0:13 Every single one of you already knows exactly

0:14 what you need to know to follow along.

0:16 So,

0:16 last episode we talked about how transistors can be used to build logic gates,

0:20 which can evaluate boolean statements.

0:22 And in boolean algebra, there are only two, binary values: true and false.

0:26 But if we only have two values,

0:27 how in the world do we represent information beyond just

0:29 these two values?

0:30 That’s where the Math comes in.

0:32 INTRO So, as we mentioned last episode,

0:43 a single binary value can be used to represent a number.

0:46 Instead of true and false,

0:48 we can call these two states 1 and 0 which is actually incredibly useful.

0:52 And if we want to represent larger things

0:54 we just need to add more binary digits.

0:55 This works exactly the same way as the

0:57 decimal numbers that we’re all familiar with.

0:59 With decimal numbers there are "only" 10 possible values a single digit can be;

1:03 0 through 9,

1:04 and to get numbers larger than 9 we just start adding more digits to the front.

1:07 We can do the same with binary.

1:09 For example, let’s take the number two hundred and sixty three.

1:12 What does this number actually represent?

1:14 Well, it means we’ve got 2 one-hundreds, 6 tens, and 3 ones.

1:18 If you add those all together, we’ve got 263.

1:21 Notice how each column has a different multiplier.

1:24 In this case, it’s 100, 10, and 1.

1:27 Each multiplier is ten times larger than the one to the right.

1:30 That's because each column has ten possible digits to work with,

1:33 0 through 9, after which

1:35 you have to carry one to the next column.

1:37 For this reason, it’s called base-ten notation,

1:39 also called decimal since deci means ten.

1:42 AND Binary works exactly the same way, it’s just base-two.

1:45 That’s because there are only two possible digits in binary– 1 and 0.

1:49 This means that each multiplier has to be two

1:51 times larger than the column to its right.

1:53 Instead of hundreds, tens, and ones, we now have fours, twos and ones.

1:57 Take for example the binary number: 101.

2:00 This means we have 1 four, 0 twos, and 1 one.

2:04 Add those all together and we’ve got the number 5 in base ten.

2:07 But to represent larger numbers, binary needs a lot more digits.

2:10 Take this number in binary 10110111.

2:12 We can convert it to decimal in the same way.

2:14 We have 1 x 128, 0 x 64, 1 x 32, 1 x 16, 0 x 8, 1 x 4, 1 x 2, and 1 x 1.

2:24 Which all adds up to 183.

2:26 Math with binary numbers isn’t hard either.

2:29 Take for example decimal addition of 183 plus 19.

2:32 First we add 3+ 9, that’s 12,

2:34 so we put 2 as the sum and carry 1 to the ten’s column.

2:37 Now we add 8 plus 1 plus the 1 we carried, thats 10, so the sum is 0 carry 1.

2:43 Finally we add 1 plus the 1 we carried, which equals 2.

2:46 So the total sum is 202.

2:48 Here’s the same sum but in binary.

2:50 Just as before, we start with the ones column.

2:52 Adding 1+1 results in 2, even in binary.

2:55 But, there is no symbol "2" so we use 10 and put 0 as our sum and carry the 1.

3:00 Just like in our decimal example.

3:02 1 plus 1, plus the 1 carried, equals 3 or 11 in binary,

3:05 so we put the sum as 1 and we

3:08 carry 1 again, and so on.

3:09 We end up with 11001010, which is the same as the number 202 in base ten.

3:14 Each of these binary digits, 1 or 0, is called a “bit”.

3:17 So in these last few examples,

3:18 we were using 8-bit numbers with their lowest value of zero

3:21 and highest value is 255, which requires all 8 bits to be set to 1.

3:26 Thats 256 different values, or 2 to the 8th power.

3:30 You might have heard of 8-bit computers, or 8-bit graphics or audio.

3:34 These were computers that did most of their operations in chunks of 8 bits.

3:38 But 256 different values isn’t a lot to work with,

3:40 so it meant things like 8-bit games

3:42 were limited to 256 different colors for their graphics.

3:46 And 8-bits is such a common size in computing, it has a special word: a byte.

3:51 A byte is 8 bits.

3:53 If you’ve got 10 bytes, it means you’ve really got 80 bits.

3:55 You’ve heard of kilobytes, megabytes, gigabytes and so on.

3:58 These prefixes denote different scales of data.

4:01 Just like one kilogram is a thousand grams, 1 kilobyte is a thousand bytes….

4:05 or really 8000 bits.

4:07 Mega is a million bytes (MB), and giga is a billion bytes (GB).

4:11 Today you might even have a hard drive that has 1 terabyte (TB) of storage.

4:15 That's 8 trillion ones and zeros.

4:17 But hold on!

4:18 That’s not always true.

4:19 In binary, a kilobyte has two to the power of 10 bytes, or 1024.

4:24 1000 is also right when talking about kilobytes,

4:26 but we should acknowledge it isn’t the only

4:28 correct definition.

4:29 You’ve probably also heard the term 32-bit

4:32 or 64-bit computers– you’re almost certainly

4:34 using one right now.

4:35 What this means is that they operate in chunks of 32 or 64 bits.

4:39 That’s a lot of bits!

4:40 The largest number you can represent with 32 bits is just under 4.3 billion.

4:45 Which is thirty-two 1's in binary.

4:47 This is why our Instagram photos are so

4:49 smooth and pretty– they are composed of millions

4:52 of colors, because computers today use 32-bit color graphics

4:56 Of course, not everything is a positive number- like my bank account in college.

5:00 So we need a way to represent positive and negative numbers.

5:03 Most computers use the first bit for the sign:

5:05 1 for negative, 0 for positive numbers, and

5:08 then use the remaining 31 bits for the number itself.

5:11 That gives us a range of roughly plus or minus two billion.

5:14 While this is a pretty big range of numbers, it’s not enough for many tasks.

5:17 There are 7 billion people on the earth,

5:20 and the US national debt is almost 20 trillion dollars after all.

5:23 This is why 64-bit numbers are useful.

5:25 The largest value a 64-bit number can represent is around 9.2 quintillion!

5:30 That’s a lot of possible numbers and will hopefully

5:32 stay above the US national debt for a while!

5:34 Most importantly, as we’ll discuss in a later episode,

5:37 computers must label locations

5:38 in their memory, known as addresses, in order to store and retrieve values.

5:43 As computer memory has grown to gigabytes and terabytes– that’s trillions of

5:46 bytes– it was necessary to have 64-bit memory addresses as well.

5:50 In addition to negative and positive numbers,

5:52 computers must deal with numbers that are

5:53 not whole numbers, like 12.7 and 3.14, or maybe even stardate: 43989.1.

6:00 These are called “floating point” numbers,

6:02 because the decimal point can float around

6:04 in the middle of number.

6:05 Several methods have been developed to represent floating point numbers.

6:08 The most common of which is the IEEE 754 standard.

6:11 And you thought historians were the only people bad at naming things!

6:14 In essence,

6:15 this standard stores decimal values sort of like scientific notation.

6:19 For example, 625.9 can be written as 0.6259 x 10^3.

6:25 There are two important numbers here: the .6259 is called the significand.

6:30 And 3 is the exponent.

6:31 In a 32-bit floating point number,

6:32 the first bit is used for the sign of the number— positive

6:35 or negative.

6:36 The next 8 bits are used to store the exponent

6:39 and the remaining 23 bits are used to store

6:41 the significand.

6:42 Ok, we’ve talked a lot about numbers,

6:44 but your name is probably composed of letters,

6:46 so it’s really useful for computers to also have a way to represent text.

6:49 However, rather than have a special form of storage for letters,

6:53 computers simply use numbers to represent letters.

6:56 The most straightforward approach might be to

6:57 simply number the letters of the alphabet:

6:59 A being 1, B being 2, C 3, and so on.

7:02 In fact, Francis Bacon, the famous English writer,

7:05 used five-bit sequences to encode

7:07 all 26 letters of the English alphabet to

7:09 send secret messages back in the 1600s.

7:11 And five bits can store 32 possible values– so that’s enough for the 26 letters,

7:16 but not enough for punctuation, digits, and upper and lower case letters.

7:20 Enter ASCII, the American Standard Code for Information Interchange.

7:24 Invented in 1963, ASCII was a 7-bit code, enough to store 128 different values.

7:29 With this expanded range,

7:30 it could encode capital letters, lowercase letters, digits

7:33 0 through 9, and symbols like the@ sign and punctuation marks.

7:37 For example, a lowercase ‘a’ is represented by the number 97,

7:40 while a capital ‘A’ is 65.

7:42 A colon is 58 and a closed parenthesis is 41.

7:45 ASCII even had a selection of special command codes,

7:48 such as a newline character to tell

7:49 the computer where to wrap a line to the next row.

7:52 In older computer systems,

7:53 the line of text would literally continue off the edge of the

7:56 screen if you didn’t include a new line character!

7:59 Because ASCII was such an early standard,

8:01 it became widely used, and critically, allowed

8:02 different computers built by different companies to exchange data.

8:06 This ability to universally exchange information is called “interoperability”.

8:10 However, it did have a major limitation:

8:12 it was really only designed for English.

8:15 Fortunately, there are 8 bits in a byte,

8:17 not 7, and it soon became popular to use codes

8:19 128 through 255, previously unused, for "national" characters.

8:25 In the US, those extra numbers were largely used to encode additional symbols,

8:29 like mathematical notation, graphical elements, and common accented characters.

8:33 On the other hand,

8:34 while the Latin characters were used universally, Russian computers used

8:37 the extra codes to encode Cyrillic characters,

8:39 and Greek computers, Greek letters, and so on.

8:42 And national character codes worked pretty well for most countries.

8:45 The problem was,

8:45 if you opened an email written in Latvian on a Turkish computer, the result

8:49 was completely incomprehensible.

8:51 And things totally broke with the rise of computing in Asia,

8:54 as languages like Chinese and Japanese

8:56 have thousands of characters.

8:58 There was no way to encode all those characters in 8-bits!

9:00 In response, each country invented multi-byte encoding schemes,

9:04 all of which were mutually incompatible.

9:06 The Japanese were so familiar with this encoding problem

9:09 that they had a special name for it:

9:11 "mojibake", which means "scrambled text".

9:13 And so it was born– Unicode– one format to rule them all.

9:17 Devised in 1992 to finally do away

9:19 with all of the different international schemes

9:21 it replaced them with one universal encoding scheme.

9:23 The most common version of Unicode uses 16

9:26 bits with space for over a million codes-

9:28 enough for every single character from every language ever used–

9:32 more than 120,000 of them in over 100 types of script

9:36 plus space for mathematical symbols and even graphical characters like Emoji.

9:40 And in the same way that ASCII defines

9:41 a scheme for encoding letters as binary numbers,

9:43 other file formats– like MP3s or GIFs– use

9:46 binary numbers to encode sounds or colors of a pixel in our photos,

9:50 movies, and music.

9:50 Most importantly, under the hood it all comes down to long sequences of bits.

9:55 Text messages, this YouTube video,

9:57 every webpage on the internet, and even your computer’s

10:00 operating system, are nothing but long sequences of 1s and 0s.

10:03 So next week,

10:04 we’ll start talking about how your computer starts manipulating those

10:07 binary sequences, for our first true taste of computation.

10:10 Thanks for watching.

10:12 See you next week.